The_Worlds_200_Hardest_Brain_Teasers_Mind_Boggling_Puzzles,_Problems

Answers 11. A single 50 percent discount is better. It is better to get a single discount of 50 percent. Do not get lured into a process that sounds superficial. In fact, it is always better to get a single discount of the sums of the successive discounts than to get the successive discounts. For example, suppose the item was originally $100. A single discount of 50 percent would give you the item at $50. Now, if I had suc- cessive discounts of 20 percent and 30 percent, the first 20 percent discount would give me $80. The second 30 percent discount on $80 would give me $56. 12. If the client died in his sleep, there would be no way of knowing what he was dreaming. 89

The World’s 200 Hardest Brain Teasers 13. Look for something that does not link smoking to cancer directly, but indirectly. That is, something that causes one to smoke and the same thing that causes cancer. So research could find that there is a certain condition that causes one to smoke and the same condition causes one to have cancer. Thus, smok- ing does not cause cancer; it is the condition that causes it. Unfortunately, that is not the case. 14. (b) 3 Do this problem in steps by starting with two socks. If you get only two socks, they could be different colors, but getting three guarantees that you will have a pair. The strategy is to realize that in order to be sure of what you get, you have to consider the worst-case scenario. The worst case is that if you reach in the drawer, you’ll get two different color socks the first two tries—let’s say a blue sock and a brown sock the first two tries. But the third try you’ll have to get either a blue or a brown sock, which will make a pair with one of the first two socks. 90

Answers 15. Thursday The key is to realize that “now” must be Friday. Look for the phrase in the problem that tells you something you can work with and use that with another part of the problem to gradually and stepwise lead to a solution. In the phrase “Two days from now will be Sunday,” you can see that now must be Friday, since two days from Friday is Sunday. Now look at the phrase “the day before yesterday.” Yesterday is Thursday since now (today) is Friday. “The day before yester- day” is Wednesday, and the “day that follows the day before yesterday” is Thursday. 16. (d) furniture : carpentry tools Put MUSIC and VIOLIN in a sentence relating the two words. Music is played on a violin by someone who knows how to play a violin, just as furniture is created by carpentry tools by a person who knows how to use carpentry tools. A crayon is not done with a drawing (drawing is done with a crayon). Note: a symphony is not played on a piano—it is played by an orchestra. 91

The World’s 200 Hardest Brain Teasers 17. (c) 10 There are two alternating sequences: 0, 1, 2, 3, 4, 5 and 0, 2, 4, 6, 8. 18. (e) John Hancock : signature Look at what is superficial and what is not. George Washington was superficially identified with the cherry tree as John Hancock was superficially identified with his flam- boyant signature. All the other people were not superficially identified in the phrase with their name. 19. (b) C minus B is always less than A Subtract the same thing from both sides. Since the sum of two sides of a triangle must be greater than the third side, A + B > C, so subtracting B from both sides, A > C – B. 92

Answers 20. (c) 19 Use a Venn diagram or write down all the possibilities. Total number of people: (a) Blonds without blue eyes (14 – 5 = 9). (b) Blue-eyed people who are not blond (8 – 5 = 3). (c) Blue-eyed blonds (5). (d) People with neither blue eyes or blond hair (2) Adding a, b, c, and d, we get 9 + 3 + 5 + 2 = 19. 21. (d) It is very tempting to just subtract the times 9:09 from 10:10, etc., and get 61 minutes, or if you’re careless get 101 minutes. However, think of the times closest to one another, 12:12 and 1:01, and you will find the difference is 49 minutes. 22. (c) 5 Draw lines to see that BD is a radius of the circle. The two diagonals of a square will have the same length. 93

The World’s 200 Hardest Brain Teasers 23. (a) 25 × 25 × 25 ×…(where there are fifty 25s) Try to find a connection between the numbers. You are compar- ing 50 × 50 × 50 × 50…(100 times) to 100 × 100 × 100 × 100… (50 times). Write 100 as 50 × 2, so now you are comparing 50 × 50 × 50 × 50…(100 times) to 50 × 2 × 50 × 2 × 50 × 2 × 50 × 2…(50 times).You can now cancel 50 × 50 × 50 × 50…50 times from both sides of the comparison. You would then get 50 × 50 × 50 × 50…(50 times) compared with 2 × 2 × 2 × 2…(50 times). You can see the left side of the comparison is 25 × 25 × 25 × 25…(50 times) times the right side. 24. Say IN means NOT, then associate EXTRIC with another word, EXTRACT. So INEXTRICABLE means not extractable—inca- pable of being extracted—or INSEPARABLE. Association is a powerful strategy for getting the meaning of words. 25. The doctor is the son’s mother. 94

Answers 26. If many people paid attention to the ad, it would be very crowded in the early hours. Of course the reason why the ad was put in was because the store did not think that so many people would read the ad. 27. (b) second Many people who get this problem wrong think of just the numbers, not the wording, assuming that if you overtake the second runner, you are first. But when you overtook the sec- ond runner, you took his or her place. Therefore you are now in second place, not first place. 28. (e) none of the above If the four daughters are those mentioned, think of who the fifth daughter could be—the fifth daughter is Bonnie herself. 95

The World’s 200 Hardest Brain Teasers 29. (e) thimble : finger Create a sentence expressing a specific relationship between the capitalized words. HELMET is worn on the HEAD to protect the head, as THIMBLE is worn on the FINGER to protect the finger. 30. If a number is an exact multiple of 9, the sum of the digits of that number is also a multiple of 9. So 1 + 2 + 3 + x = 9 and x = 3. 31. The knight could tear up the paper he picks and offer the other one to the king. He could then tell the king that since the untorn paper reads “Death,” the one torn must have read, “Marriage.” 96

Answers 32. From first to last: John, Mary, Sarah, Barry The easy way to find the answer is to draw the situation with labels for the people’s names. (1) Sarah is between Barry and Mary: B S M or M S B. (2) John is directly in front of Mary: J M (3) Mary is in front of two other people: M S B So we get from (2) and (3): J M S B 33. One-half the surface area of a ball is greater. Blow up the circle like a balloon to get the ball. One half the surface area of the ball is greater than the area of the circle because the circle is being expanded by being blown up. 34. Write a three-digit number with h hundreds digit, t tens digit, and u units digit as 100h + 10t + u. You will find that A = 1, B = 0 and C = 9. We have 100A + 10B + A – 10C – A = 10A + B. This gives us 90A + 9B = 10C or 10A + B = (10/9)C. 97

A The World’s 200 Hardest Brain Teasers The only way A and BDcan be integers is if C = 9. That makes 10A + B = 10. The only way this is possible with A and B as integers (from 0 to 9) is if A = 1Eand B = 0. 20° 10° 60° 70° B C 35. (b) m Letters are arranged in pairs, backward skipping every third letter: m n (o) p q (r) s t (u) v w (x) y z. 36. 5 Draw a perpendicular line to make a rectangle in the figure and you will also have a 3-4-5 triangle. 5 4 45 53 8 A powerful strategy in geometry is to draw extra lines to get more information from the problem. You may not remember this, but when you first took geometry, one of the first proofs that the teacher probably showed was “if two sides of a tri- angle are equal, then the base angles are equal.” The teacher drew a perpendicular line to start the proof—unfortunately 98

Answers the teacher probably did not tell you that the reason she did this was because when you draw something you get more information and are able to start solving the problem. A natu- ral phenomenon! So in this problem, draw a line to make a rectangle in the figure and you will find that you will have a 3-4-5 triangle. x = 5. 37. (c) 40 percent Translate words to math. Percent can be translated to 1/100, of to X, what to x, is to =. A = [250/100](B). x/100 (A) = B. Substitute: x/100[(250/100)]B = B. Cancel B: x/100[250/100]  =  1. 250x/10000  =  1 so x = 10,000/250 = 40. 99

The World’s 200 Hardest Brain Teasers 38. The average rate is less than the average of the rates. Average rate = 2AB/(A + B). Average of the rates = (A + B)/2. Detailed solution: From the formula of Rate × Time = Distance, calling the time for the car to go uphill, t, and the time for the car to go down- hill, T, with D being the distance downhill and uphill, (1) A × t = D and (2) B × T = D. Now, Average rate = Total distance/Total time. So, (3) Average rate = 2D/(t + T). From (1) we have (4) t = D/A and from (2) we have (5) T = D/B Substituting from (4) and (5) in (3), we get (6) Average rate = 2D/(D/A + D/B) This becomes (7) Average rate = 2AB/(A + B) But the average of the rates is just (A + B)/2. So we ask, is 100

Answers 2AB/(A+B) greater, equal to, or less than (A+B)/2? So we compare 2AB/(A+B) with (A+B)/2. Put 2AB/(A+B) under a Column A and (A+B)/2 under a Column B and let’s compare the Columns: Column A Column B 2AB A+B A + B 2 Now multiply both columns by 2 and then by (A + B). We get Column A Column B 4AB (A + B) (A + B) This becomes Column B A × A + 2AB + B × B Column A 4AB Now subtract 4AB from both columns. We get: Column A Column B 0 A × A – 2AB + B × B But A × A – 2AB + B × B = (A – B)(A – B) so we get: Column A Column B (A – B) (A – B) 101

The World’s 200 Hardest Brain Teasers Since we are told that A and B are different rates, A – B cannot be 0, and so (A – B)(A – B) is always greater than 0 whether A is greater than B or less than B. So Column B is greater than Column A and the original quantity in Column B, the average of the rates, is greater than the original quan- tity in Column A, the average rate. 39. e, a, c, b, d 40. (e) 13 Almost everyone who tries this problem gets it wrong! To get the right answer, you have to visualize something that is not straightforward. When this problem first appeared in the Washington Post and other newspapers throughout the country in 1995, forty percent of people answered (a) 9 because they just added the small triangles in the figure, ABC, BDE, BEC, etc. Twenty- five percent of people answered (b)10 because they added the small triangles in the figure and also added the big triangle AGJ. Eight percent of people answered (c) 11 because they added the small triangles, the big triangle AGJ, and triangle 102

E10° Answ20e°rs 60° 70° ADF. Sixteen percentBchose (f) none of theCabove, many of whom merely guessed at an answer without really working it out. Seven percent of people chose the correct answer, (e) 13, because they added the small triangles, the big triangle, triangle ADF, and the trian5gles BGI and CHJ. Note: Four percent4of people chosex(d) 12—they were almost correct. They added the small triangles, the big triangle, and triangle ADF but then only8added the triangle BGI and not also CHJ. A BC D EF G HI J 41. When the bird flies, it pushes down on the air, which pushes down on the scale. The scale reads the same. 103

The World’s 200 Hardest Brain Teasers 42. (e) cannot be determined because this is an ambiguous question How can you overtake the person who is last? 43. (b) 202/962 is greater. Add 208/962 to both quantities and compare. 44. (e) termite : house Create a sentence expressing a specific relationship between the capitalized words. MOTH is a living thing that destroys CLOTHING, as termite is a living thing that destroys a house. 104

Answers 45. (a) No good person lives to an old age and (d) All bad people do not die young. “Only the good die young” means that those who are not good do not die young and no good person does not die young. Choice (c) Only bad people do not die young would have been correct if all people were either good or bad. Some may be neither. 46. (e) Phil could give Harry $20 and could give Sam $10. This could really give you a headache if you don’t represent what’s given in a table. HSP Harry owes +$30 –$30 Harry is ahead $30 and Sam $30 Sam is behind $30 Sam owes Phil +$20 –$20 Sam is ahead $20 and $20 Phil is behind $20 Phil owes –$50 +$50 Phil is ahead $50 and Harry $50 Harry is behind $50 –$20 –$10 +$30 Phil must give Harry Total $20 and Sam $10 to make totals 0 105

The World’s 200 Hardest Brain Teasers 47. (d) I and II only If nobody loves nobody, then it is like saying nobody loves zero people. So they must love more than zero people. Then somebody loves somebody and everybody loves somebody. 48. (e) 2 × 6 × 36 When you have to choose something different from the rest of other choices, look for either something all the other choices have that the correct one doesn’t, or something that the correct choice has that the other choices don’t have. So don’t multiply: Notice that all choices are divisible by 5 ex- cept choice (e), 2 × 6 × 36. Also, (e) 2 × 6 × 36 is the only choice that is divisible by 9. 106

Answers 49. (c) 37 percent The equivalent single discount is less than the sum of the discounts. Start with $100. A 30 percent discount gives you a $70 price. A 10 percent discount on $70 gives you a price of $63. $100 – $63 = $37 discount, which is equivalent to a 37 percent discount on $100. 50. 25° To find the solution, label all the angles with angle BEC = x. Let ∠AEB = y, ∠BEC = x, and ∠CED = z. Since ∠AED = 90, (1) x + y + z = 90. Since ∠BED = 40, (2) x + z = 40. Since ∠AEC  =  75, (3) y  +  x  =  75. Subtracting Equation (2) from Equation (1) we get y = 50. Substituting y = 50 in Equation (3) we get x = 25. AB C E D 107

The World’s 200 Hardest Brain Teasers 51. 999 × 1,001 is just 1 less than 1,000 × 1,000. One of the key strategies in mathematics is to write what is presented in a different form: Write 999 × 1,001 as (1,000  –  1)  ×  (1000  +  1) and you’ll find that’s equal to 1,000 × 1,000 – 1, since (a –1) × (a +1) = a × a – 1. 52. 276 Each person can shake hands with 23 other people, so we have 24 × 23 possible combinations. But because it takes two people to make a handshake, we must divide this prod- uct by 2, so we get (24 × 23)/2, or 276. 53. DRAW is the only verb on the list with a past tense that does not rhyme with OUGHT. The strategy is to try to see something curious in the words. Notice something that many of the words have in common. Most of them have a past tense that has a different struc- ture or sound from the word. BRING—not “BRINGED” but BROUGHT, BUY—not “BUYED” but BOUGHT, 108

Answers CATCH—not “CATCHED” but CAUGHT, etc. And all of these past tense words rhyme with OUGHT, except “DREW,” which is the past tense of DRAW. So DREW is the odd man out! 54. (b) There is one right angle in the first set for each figure and two right angles in each figure in the second set. So the missing figure must have two right angles. 55. There are actually several correct answers to this problem. Here are some of them: A-E-T makes EAT, TEA, and ATE; A-T-R makes ART, RAT, and TAR; O-T-P makes OPT, TOP, and POT; A-P-T makes APT, PAT, and TAP; R-E-A makes EAR, ARE, and ERA. The thing to do is to figure out how to start a problem like this. What do you do first? First, just try to make a three-letter 109

The World’s 200 Hardest Brain Teasers word—any three-letter word, like TIN. Using those letters, T-I-N, can you make another word? Start with T: TNI does not make a word. Now methodically use I as the first letter: INT and ITN do not make a word. Now start with N: NIT and NTI do not make a word—so T-I-N does not work. Don’t get discouraged. Try another three-letter word: E-A-T. EAT is a word. Start with A now: ATE is a word. Now start with T: TEA is a word. Other correct answers are OPT, TOP, POT, and ART, RAT, and TAR. 56. The only number that meets these requirements is 735. 57. The only English word that can be formed by two consecutive three-letter abbreviations for months is DECOCT, which means to extract the flavor by boiling or to steep in hot water. 110

Answers 58. AD × CD is greater than h × AB. To solve, label the sides of the parallelogram with letters like a and b, then cancel the like quantities. A a B bh b C Da h × AB = h × a and AD × CD = b × a We are then comparing h × a with b × a. Since the a is the same in each expression, we are really comparing h with b, and h is le3ss tha7n b because b is the largest side (the hypot- enuse) of7a right16triangle. Thus the original h × AB is less than AD × CD. 16 × 3 7 × 7 48 49 48 < 49 59. The long hand passes the short hand only eleven times. The long hand passes the short hand at noon, and between the times 1 pm and 2 pm, 2 pm and 3 pm, 3 pm and 4 pm, 4 pm and 5 pm, 5 pm and 6 pm, 6 pm and 7 pm, 7 pm and 8 pm, 8 pm and 9 pm, 9 pm and 10 pm, 10 pm and 11 pm a total of eleven times. Note that the long hand does not pass the short hand between 11 pm and 12 am, it just meets the short hand at 12 am. 111

The World’s 200 Hardest Brain Teasers 60. One word is PASSIVE. When the prefix IM is placed before that word, creating the word IMPASSIVE, that word has the same meaning. Other pairs are RAVEL-UNRAVEL, FLAMMABLE- INFLAMMABLE, and VALUABLE-INVALUABLE. 61. I know that you probably were trying to figure out the sides and the relationship of all the lengths AC, AB, BD, and CE. But what you probably didn’t think of is that the area of triangle ABC is represented as CE × AB/2 but also as BD × AC/2. So (1) CE × AB = BD × AC. Now if AC is greater than AB (given), in order for (1) to be true, CE must be greater than BD! 62. 24 (3 × 4 × 2) 63. (c) 4,100 1,000 + 40 + 1,000 + 30 + 1,000 + 20 + 1,000 + 10 = 4,100. 112

Answers 64. (d) p Look for a relationship between the letters. Between a and b there are no letters. Between b and d there is one letter (c), Between d and g there are two letters (e, f). Between g and k there are three letters (h, i, j). To continue the pattern, skip four letters (l, m, n, o). The next letter in the sequence is p. 65. 666 The integers between 0 and 1000 that are exactly divisible by 3 are 3, 6, 9, 12, 15, 18… You can see that there are 999/3  =  333 integers between 0 and 1000 that are exactly divisible by 3. The integers between 0 and 1000 are 1, 2, 3… 999. This is a total of 999 – 1 + 1 = 999 integers. Thus 999 – 333 = 666. 113

The World’s 200 Hardest Brain Teasers 66. 100 Don’t be careless. Watch what you are dividing. You’re not dividing by 2, you are dividing by 1/2. So you have to know your basic skills. Diving by 1/2 is like multiplying by two. So the answer is (40 × 2) + 20 = 80 + 20 = 100. 67. (c) trout : fish You may have thought the answer was (a), but a TURTLE is a creature, which is a type of REPTILE, as trout is a creature, which is a type of fish. 68. 1. please 2. ground 3. around 4. knight 5. fright 6. danger 114

Answers Here are some more: 7. trivet 8. strove 9. shovel 10. slight 11. mother 12. craven 69. c, e, a, d, b 70. (c) U In set I, all three symbols are represented by straight lines. In set II, the two symbols are represented by straight and curved lines. 71. (b) 2 One of them is not 2, it is 1. But the other is 2. 115

The World’s 200 Hardest Brain Teasers 72. The triangle shown has a base of 6 and sides 2 and 4. Since 2 + 4 = 6, the triangle has collapsed into a straight line. On this line, the line segment labeled x must be 1. 73. 28 Lowest average score = [21 × 80 + (60 – 21) × 0]/60 = 28 74. 640 (9 × 6 × 3)x = 6 × 12 × 4 × 12 × 2.5 × 12; x = 640 116

A a B bh b C Answers Da 75. 7/16 is greater. Multiply 16 × 3 and 7 × 7 in the fractions. 48 < 49 so 3/7 < 7/16. 37 7 16 16 × 3 7 × 7 48 49 48 < 49 76. You can simplify the problem by multiplying the numerator and the denominator by the same number. This process will not change the value of the original fraction. So, multiply both nu- merator and denominator by 4 and you get 4,444 divided by 100, which is 4,444 percent. 77. Roots are a powerful tool in figuring out the meaning of words. The root MAN means “hand.” The root MIT means “to send.” So MANUMIT means to send by hand or set free. 117

The World’s 200 Hardest Brain Teasers 78. (a) n There are actually two sequences here: a (b) c (d) e (f) g (h) i…and z (y) (x) w (v) (u) t (s) (r) q (p) (o) n… 79. (e) 7 If people are denoted by a, b, c, you have committees a, b, c, ab, ac, bc, and abc. 80. (a) 11/20 81. 21 Represent b = a + 2, c = a + 4, so a + b + c = 3a + 6 = 57. If 3a = 51, a = 17, so c = a + 4 = 21. 118

Answers 82. 400 Let n be the number of original shelves. Then 80n = 50(n + 3). 30n = 150; n = 5. 80 × 5 = 400. 83. An hourglass 84. No water—it would be all strained out. 85. 35 × 65 is greater. Divide both products by 65 × 34. You then only have to compare 35/34 with 66/65. Since 35/34 = 1 1/34 and 66/65 = 1 1/65, the first product in the question is greater. 86. (f) 36 The final result must be divisible by 9. It is very interest- ing that when you add the digits of a number and subtract 119

The World’s 200 Hardest Brain Teasers that from the original number you get a number that is a perfect multiple of 9. Here’s the proof: Represent the two- digit number as 10t + u where t is the tens digit and u is the units digit. Adding the digits, we get t + u. Subtracting from the original number, we get 10t + u – t – u = 9t. Since t is an integer, the only choice that fits is (f) 36. What’s more in- teresting is that any number that is a multiple of 9 has digits that add up to a multiple of 9. Here’s a great parlor trick: Have someone choose a number. Then have that person add the digits. Then have the person subtract that result from the original number to get a final result. Have the person now cross out one of the digits in the final answer. You will be able to tell the person what digit he crossed out if he tells you the remaining digit. For example, he starts with 23. 2 + 3 = 5. 23 – 5 = 18. He crosses off the 8. He is left with 1. You will be able to tell the digit crossed out. Why? Any number that is a multiple of 9 has digits that add up to a multiple of 9. So whatever number you crossed out, the crossed-out number must be 9 minus the number left! 120

Answers 87. The mind may give you the pronunciation of DOUS as “doos.” Don’t assume that is the correct pronunciation. Realizing that, you may come up with the words stupendous, tremendous, or horrendous. 88. (e) a country MEANYRG spells GERMANY. 89. (d) Spaceship Look for similarities in the messages: Avion occurs in (1) and (2) and so does Serious. So Avion must mean Serious. Balcon appears in (1) and (3) and so does Fumes. So Balcon must mean Fumes. So looking at (1), Sondor must mean Spaceship. 121

The World’s 200 Hardest Brain Teasers 90. (d) 240 Do this in steps: If eight people can wash 50 cars in 60 hours, then four people can wash 50 cars in 120 hours. It would take twice as much time to wash, since there are half as many people to wash. Thus, it would take four people 240 hours, twice as long, to wash 100 cars, since there are twice as many cars now. 91. (c) cannot be determined as true or false Translate to something you know. For example, let Blips be Students and Plips be Teachers. And let Jips be Actors. So “Some students are actors” is neither false nor true. 92. (c) 99 This is tricky. If Matt is the fiftieth fastest runner, he would be number 50 in the sequence 1, 2, 3…50. To be the fiftieth slowest, he’d have to be number 50 in the sequence 50, 51, 52…99, since there are fifty numbers from 50 to 99 inclusive. 122

Answers 93. (d) None of the above can be proved true. Write P > N and D < P. This is the same as P > N and P > D. You cannot determine the relationship of D and N. For example, if P is be 6', N can be 5', and D can be 5.5' or N can be 5' and D can be 5' or N can be 5.5' and D can be 5.5'. 94. (d) 24 Write the number of fish Harry caught as H and the num- ber of fish Sam caught as S. Then translate to math: (1) H = 3S. So, (2) H + S = 32. Substitute H = 3S in (2) and we get 3S + S = 32. 4S = 32, S = 8. So, H = 3S = 24. 95. (c) 10 Draw a diagram to illustrate the situation. 5 4 03 34 5 The distance between the two runners is 10. 123

The World’s 200 Hardest Brain Teasers 96. MAINE is the shortest name of a state that shares a letter in common with each of the other fifty states 97. 24 Start with the first person, a, at the left side in a corner posi- tion. Then figure out the combinations you can have (ab- cd or ac-bd, etc.) Since a is in the left corner, you can play around with combinations of bcd for the remaining positions. There are 3 × 2 = 6 of them. Now b, c, or d can be in the left corner also, so there are 6 + 6 + 6 more combinations. That gives us a total of 24. 98. No Watch the wording. A widow has lost her husband. His wid- ow would be his present wife, but he is dead, so that’s impos- sible. If the wording said “a widow’s sister,” then it would be possible, since he would not be her dead husband. 124

Answers 99. 9 You may have thought of subtracting 9 from 17, giving you 8. But the question says “all but 9 die.” That means you have 9 left! 100. (d) 201 Place 11 cartons with the 40\" side along the back of the room, and repeat in front of this row so you have 15 rows deep of 11 cartons wide (165 cartons). In the remaining space, place two cartons with their 48\" side against the back and make 18 rows of these (36 cartons). This gives 165 + 36 = 201 refrigerators stored. You can also get 201 fridges in by changing the numbers so you have 15 rows of 5 cartons and 18 rows of 7 cartons. 101. Quantity B Understand the difference between “average rate” and the “average of the rates.” Then cross-multiply, then subtract 4ab from both quantities. 125

The World’s 200 Hardest Brain Teasers Detailed explanation: The average rate of a car going uphill at a miles per hour and downhill at b miles per hour the same distance is just 2ab/(a + b). Proof: Call the distance d. Call the time it takes for the car to go uphill t. Call the time it takes for the car to go downhill the same distance, d, T. Because Rate × Time = Distance, (1) a × t = d (2) b × T = d The average rate is the total distance divided by the total time. That is 2d/(t + T). Using (1), we get t = d/a and (2), T = d/b. So 2d/(t +T) = 2d/(d/a + d/b), which can be seen to be just 2ab/(a + b). The average of the rates a and b is just (a + b)/2. So Quantity A is just 2ab/(a + b) and Quantity B is just (a + b)/2. Multiply both quantities by 2. You get Quantity A: 4ab/(a + b) Quantity B: a + b Now multiply both quantities by (a + b). You get Quantity A: 4ab 126

Answers Quantity B: (a + b) × (a + b) = a2 + b2 + 2 × b × a Now subtract 4ab from both quantities. You get Quantity A: 0 Quantity B: a2 – 2 × b × a + b2 = (a – b) × (a – b) Since a is not equal to b, (a – b) × (a – b) is always greater than 0. Thus Quantity A is less than Quantity B and so the original Quantity A [2ab/(a + b)] is less than the original Quantity B [(a + b)/2]. Thus, the average rate of the car going uphill at a miles per hour and downhill at b miles per hour is less than the average of the rates a and b. 102. (b) Mike is a child. 103. Small Adding er gives you smaller. You can also do this with meter. Adding milli makes the word millimeter, which is smaller. 127

The World’s 200 Hardest Brain Teasers 104. Yes Don’t get lured by the fact that the fourth of July is a holiday celebrated only in the United States. It doesn’t say celebrated or holiday in the question. So of course they have a fourth of July (July 4) in England, since that’s also a day in their calendar. 105. One Analyze what a birthday is. Don’t get lured into the word average. It’s a single day that someone was born. So the aver- age man (or any man) has only one birthday, his date of birth. 106. 6 You may have thought the answer was “three outs.” But there are two teams that play in an inning. So there are six outs. 128

Answers 107. See what you have been given to work with in answering the question. Since the problem doesn’t give any indication of the butcher’s weight, the word weigh must refer to something else. The butcher weighs meat  !  108. (e) The ? must have only one set of parallel lines. 109. 12 stamps You may have multiplied 2 × 12 to get 24. But a dozen stamps means 12 stamps no matter how much each stamp is worth! 129

The World’s 200 Hardest Brain Teasers 110. 60 minutes Many of you probably think that I should multiply 3 pills by 30 minutes to give me 90 minutes. But think of how you take the pills. I take the first pill, and in a half-hour I take the second pill, and then in the next half-hour I take the third pill. So the number of minutes that pass from the first to the third pill is 60 minutes. 111. Ten numbers Start with the lowest number: 4,567. The next number is 4,568, then 4,569; 4,578; 4,579; 4,589; 4,678; 4,679; 4,689; 4,789. So there are ten numbers. 112. 1 Note that the numbers in the sequence are integers multiplied by themselves three times, that is 125 = 5 × 5 × 5; 64 = 4 × 4 × 4; 27 = 3 × 3 × 3; 8 = 2 × 2 × 2. So the next number would be 1. 130

Answers 113. (e) 45 If John = 10, then his mother is 40. When John is 15, he is 5 years older. So, his mother must be 5 years older, and then is 45. 114. (a) 6 You need to use the formula Rate × Time = Distance, that is r × t = d. For the uphill trip, r (uphill) × t (uphill) = 1 mile For the downhill trip, r (downhill) × t (downhill) = 1 mile The average rate of the car is the total distance traveled (2 miles) divided by the total time traveled, t (uphill) + t (downhill). Since the average rate was given as 20 mph, 20 = 2 divided by [t (uphill) + t (downhill)]. Thus, we get 10 = t (uphill) + t (downhill). 131

The World’s 200 Hardest Brain Teasers 115. (d) You can’t do anything. Since you’re going from a very high speed to 0 (when the elevator hits the ground) the force is tremendous (from phys- ics, force is proportional to the acceleration). You can’t jump up because you’d be going from a very high speed to a speed of less than 0 and that would also require a tremendous force. The force is what gets you, so there’s nothing you can do! 116. 46 × 767 × 72 is less than 767 × 46 × 74. Cancel 46 and 767 (common numbers) from both products. 117. The house that is painted yellow has just been sold. The house, which is painted yellow, has just been sold. In (1), “The house” refers to one of a number of houses. In (2), “The house” does not refer to one of a number of houses. 132

Answers 118. Seven Draw a line in the circle and place nine people on the line. Choose two people on the line and draw another circle inside the first so that those two people are on the circumference of that circle. Thus you would have to move the rest of the people (7 people) to place them also on the circumference of that circle. 119. 1 is greater. Multiply 1 by 99/77 and compare that with 77/99. 120. (b) replace…important Use key words: while, is not able, it is still able. 121. It confuses people with playing styles. 133

The World’s 200 Hardest Brain Teasers 122. (e) freezing : spoilage A VACCINATION is used to prevent a DISEASE as freezing is used to prevent spoilage. Note that aspirin is not used to prevent a headache; it can be used to reduce the symptoms of a headache and also as a regiment to prevent against heart attacks. 123. 7:04 am The clock gains 3 minutes every day, or 1 minute every 8 hours. Between 11 pm Sunday night and Tuesday morning 7 am is 24 + 8 hours. So the clock gains 4 minutes. 124. Average speed is total distance divided by total time. So for the first half, 25 = 50/time of first half. Thus the time for first half is 2 hours. For Phil to average 50 miles per hour for the whole trip, we would get 50 = 100/time for the whole trip. Time for the whole trip then is 2 hours. That would mean that Phil must go an infinite speed the second half of the trip to go 50 miles in 0 hours. 134

Answers 125. (c) 3 vs. 3 Label the 9 coins a, b, c, d, e, f, g, h, i. First weigh abc with ghi. If they balance, then heavy coin is d, e, or f. So weigh d with e. If they balance, then the heavy coin is f. If they don’t balance, then whichever coin tips the scale is the heavy one. Suppose abc and ghi do not balance. Then whichever side tips the scale has the heavier coin. Suppose in this case the side with ghi has the heavy coin. Then, in the procedure done before, weigh g with h. If they balance, then i is heavy. If they don’t balance, then the coin that tips the scale downward is heavy. Here is a more sophisticated solution: Number the coins 1 through 9. In the first weighing, place coins 1, 2, and 3 on the left side of the scale, and place 4, 5, and 6 on the right side. Leave 7, 8 and 9 on the side. For the second weighing, regardless of the results of the first, place coins 1, 4, and 7 on the left, and place 2, 5, and 8 on the right. The following chart demonstrates how to determine which is the heaviest coin depending on these results. In this chart, the results displayed list which side of the balance scale was heavier. 135

The World’s 200 Hardest Brain Teasers Heavy Results of First Results of Second Coin Weighing Weighing 1, 2, 3, vs. 4, 5, 6 1, 4, 7, vs. 2, 5, 8 1 Left Left 2 Left Right 3 Left Equal 4 Right Left 5 Right Right 6 Right Equal 7 Equal Left 8 Equal Right 9 Equal Equal 126. He or she should open the remaining door. The probability would be 2/3, not 1/2! Many people have thought that there is a 1/2 chance of having the car behind the door the contestant chose since there are two doors. However, because you have chosen a door first, out of three, there is a 1/3 probability there is a car behind it, and there would be a 2/3 probability any other door has the car behind it. So if you chose two of the other doors, you would have a 2/3 probability of the car. And if one door was eliminated 136

Answers by the host, since that door had the goat, if you switched doors, then it would be in effect as if you were choosing two doors, which would give you the 2/3 probability that the car was behind the remaining door. Another explanation: Interestingly enough, the answer is that the contestant should switch doors, because there is actually a 2/3 chance of winning the car by switching, while there is only a 1/3 chance of win- ning the car if the contestant opens his or her original door! This is how most people’s minds approach the problem: The host eliminates one door with the goat, so there is a goat or a car behind the contestant’s door. So the probability is 1/2. And it doesn’t matter whether the contestant opens his or her original door or changes doors. But let’s see what is really happening. The probability of having a car behind the contes- tant’s door when he or she originally chooses the door is 1/3, since there are three doors and only one door that has a car behind it. And there is a 2/3 probability that the car is behind one of the other two doors (since there are two ways a car can be behind one of the remaining doors [car-goat; goat-car] out of a total of three possibilities: car-goat; goat-car; goat- goat). Now, if the host eliminates a door, there would still be the 1/3 probability that the car was behind the contestant’s original door—so once a door is eliminated, there is a 2/3 probability the car was behind the remaining door. 137

The World’s 200 Hardest Brain Teasers I have pondered the Let’s Make a Deal problem for some time, and I came to the conclusion that it is 2/3 when I took a more dramatic variation of the problem. Suppose I had 100 doors and 1 of them had a car behind it. The rest had goats. If I chose one door, there would no doubt be a 1/100 chance that there was a car behind it. So the remaining doors combined would have a 99/100 chance of the car. Now no matter what was done to the other doors, there would still be a 1/100 chance of the car’s being behind the door that was chosen. So if the host knew that 98 of the doors did not have the car, and then eliminated those doors, the remaining door then must have had a 99/100 chance that there was a car behind it. Suppose you had 100 doors. The chance of the car’s being be- hind the contestant’s chosen door is 1/100, obviously a very slim chance that won’t change even when the host eliminates 98 doors. When the doors are eliminated, and all but two are left, the contestant still has chosen a door with a very, very slim chance the car is behind it, namely 1/100. Because the host knows where the car is (and where the goats are), and because 98 of the goats were eliminated, the remaining door has a (98 + 1)/100 chance there is a car behind it. Note that initial reasoning may indicate that when there is only one door left, since 98 doors have been eliminated, the total number of possibilities for what’s behind the contestant’s 138